Subject: CCL:Anharmonicity (Summary)

From: "Jens Spanget-Larsen" <[email-masked]>
Date: Fri, 22 Nov 1996 14:10:19 +0100

CCL:

On Friday, 15 Nov 1996, I posted the following question: 
 
> In a study of molecular vibrational structure I have encountered a 
> normal mode with a "symmetrical" anharmonic potential V of the form: 
> 
>  V(Q) = a*Q^2 + b*Q^4
> 
> i.e., the potential V as a function of the normal coordinate Q can be 
> approximated by a fourth order polynomial in Q, containing only the 
> second and the fourth power.  What are the solutions to the 
> vibrational eigenvalue problem with this potential function?  I would 
> be most grateful for information on literature that deals with this 
> problem.

I immediately received a number of useful replies, that are 
summarized below.  Thank you to all that replied!

Yours, Jens >--<

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Date sent:        Fri, 15 Nov 96 10:21:42 PST
From:             [email-masked] (Herb Strauss)

Please look at papers by D. Harris et al, J Laane, et al, W D Gwinn et
el and H. L. Strauss et al, mostly in the sixties.
Herb Strauss
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From:             "MARTIN L. SAGE X-2713" <[email-masked]>
Organization:      Syracuse University
Date sent:         Fri, 15 Nov 1996 14:52:34 EST 

Dear Jens,
     I wrote a paper "An Algebraic Treatment of Quantum Vibrations
Using REDUCE" , J. Symbolic Computation (1988)5, 377-384 which treats
this potential.  If REDUCE isn't avaailable I would consider sending
more information.
                                Sincerely,
                                Martin
MARTIN L. SAGE
DEPARTMENT OF CHEMISTRY
SYRACUSE UNIVERSITY
CST 1-014
SYRACUSE, NY 13244-4100
USA
[email-masked]
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Date sent:        Fri, 15 Nov 1996 11:21:21 -0500
From:             Sam Abrash <[email-masked]>

This is the potential for an anharmonic bending motion, the solution
can be found in Wilson, Decius and Cross, Molecular Vibrations or any
other standard text on vibrational spectroscopy.

Samuel A. Abrash
Department of Chemistry
University of Richmond
Richmond, VA  23173
Phone: (804) 289-8248
Fax: (804) 289-8482

UNTIL DECEMBER:  [email-masked]

DECEMBER and AFTER:

[email-masked]
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Date sent:        Fri, 15 Nov 1996 13:51:20 -0500
From:             Russ Johnson <[email-masked]>

Hello!
   There is a nice paper on that potential form by
Jaan Laane in 
Applied Spectroscopy Volume 24 Number 1 page 73 (1970). 
He has solved it using harmonic oscillator 
wavefunctions, and had provided tables
from which the lowest sixteen eigenvalues can be
interpolated.

Russ Johnson

Dr. Russell D. Johnson III
Research Chemist
Physical and Chemical Properties Division
National Institute of Standards and Technology
Gaithersburg, MD 20899
voice 301+975-2513     fax  301+975-3670
email: [email-masked]
###################################################################
Date sent:        Fri, 15 Nov 1996 14:31:57 -0500
From:             Brian Teppen <[email-masked]>

Hello, Jens:

The eigenvalues depend on coefficients a and b of course. If a is
negative and has the right proportionality to b, then you get a
symmetric, double-well potential. The force constant (second
derivative) can give the same value when evaluated at either minimum.
Each well will be somewhat anharmonic. I have used such a potential
(paper in review) to model the angle-bending of octahedral systems, so
that there will be minima at both 90 and 180 degrees but not zero
degrees. I used a=(-1.234)*b to get the two, symmetric minima.

Another physical example (where I was first exposed to the idea)
would be the inversion of NH3.

Best wishes,
Brian J. Teppen                                    [email-masked]
Advanced Analytical Center for Environmental Sciences Savannah River
Ecology Laboratory University of Georgia Drawer E Aiken, SC 29802

phone:803-725-8157                    fax:803-725-3309
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From:             TOPPER ROBERT <[email-masked]>
Subject:          Re: CCL:Anharmonicity
Well, don't quote me on this but I don't think that an exact
analytical solution exists. There are only a few oscillator problems
which yield analytical solutions; the most notable being the harmonic
oscillator and the Morse oscillator. There are a couple of others too.

Of course, you could use perturbation theory to
form an approximate solution, treating the Hamiltonian 
H = T(p) + k*Q^2/2 as the zeroth-order part (for which the solution is
known) and bQ^4 as the perturbation. Within this approximation the
energy is given to first order by E = E^0_n + E'_n where E^0_n = hbar
* omega * (n+0.5) with omega = sqrt (k / m ) (m the mass of the
oscillator) and E'_n = b * < n | Q^4 | n > , i.e. the expectation of
the Q^4 operator within the nth quantum state. This may or may not
have a simple closed-form solution for all n either, but could be
easily done for the case where n=0 (the ground state). I believe that
the result will be on the order of hbar^2 if memory serves.

I'll leave the rest of the algebra for you to work through; 
have a look at Chapter 7 of Donald MacQuarrie's "Quantum Chemistry" or
any other comparable quantum chemistry textbook.

Hope this helps,
robert

**********************************************************************
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The Cooper Union                      phone:  (212) 353-4378 51 
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receive full scholarships.

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Date sent:        15 Nov 1996 17:47:34 -0500
From:             "John Beckerle"
                  <[email-masked]>

        Reply to:   RE>CCL:Anharmonicity

I would be very interested to see the responses to your question.
Thanks.
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From:             [email-masked] (Don Secrest)
Date sent:        Tue, 19 Nov 1996 11:58:26 -0600 (CST)

Dear Dr. Spanget-Larsen:
 Many years ago I tried to solve the problem with the potential
X**4 only.  This seemed so easy, but I was unable to solve it after
much work.  I talked to a Professor in the math department and he was
quite sure it couldn't be solved in closed form.  So I settled for a
numerical solution and developed a method for solving the problem
X**(2*n).  It is published in J. Chem. Phys. 37,830(1962).
 I am still not convinced that this problem is unsolvable in
closed form, however.  If you manage to solve your problem I would be
very interested in seeing your solution.
   Sincerely yours
      Don Secrest
   [email-masked]
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
JENS SPANGET-LARSEN  
Department of Chemistry      Phone:  +45 46757781 + 2710
Roskilde University (RUC)    Fax:    +45 46757721 
P.O.Box 260                  E-Mail: [email-masked]
DK-4000 Roskilde, Denmark    http://frederik.ruc.dk/dis/chem/psos
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